Let $t \in \{2,8,10,12,14,16,18\}$ and $n=31s+t\geq 14$, $d_{a}(n,5)$ and $d_{l}(n,5)$ be distances of binary $[n,5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n,5,d_{a}(n,5)]$ optimal linear code is not an LCD code, there is an $[n,5,d_{l}(n,5)]=[n,5,d_{a}(n,5)-1]$ optimal LCD code if $t\neq 16$, and an optimal $[n,5,d_{l}(n,5)]$ optimal LCD code has $d_{l}(n,5)=16s+6=d_{a}(n,5)-2$ for $t=16$. Combined with known results on optimal LCD code, $d_{l}(n,5)$ of all $[n,5]$ LCD codes are completely determined.

翻译：令$t \in \{2,8,10,12,14,16,18\}$且$n=31s+t\geq 14$，$d_{a}(n,5)$和$d_{l}(n,5)$分别表示二元$[n,5]$最优线性码和最优线性互补对偶(LCD)码的距离。我们证明：当$t\neq 16$时，$[n,5,d_{a}(n,5)]$最优线性码不是LCD码，但存在$[n,5,d_{l}(n,5)]=[n,5,d_{a}(n,5)-1]$的最优LCD码；当$t=16$时，最优$[n,5,d_{l}(n,5)]$ LCD码满足$d_{l}(n,5)=16s+6=d_{a}(n,5)-2$。结合已知的最优LCD码结果，所有$[n,5]$ LCD码的$d_{l}(n,5)$被完全确定。